Rates of convergence in the multivariate weak invariance principle for nonuniformly hyperbolic maps

We obtain rates of convergence in the weak invariance principle for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper R Superscript d"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">\mathbb {R}^d</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -valued Hölder observables over nonuniformly hyperbolic maps. In particular, for maps modelled by a Young tower with superpolynomial tails (e.g. the Sinai billiard map, and Axiom A diffeomorphisms) we obtain a rate of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper O left-parenthesis n Superscript negative r Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mi>n</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo> − </mml:mo> <mml:mi>r</mml:mi> </mml:mrow> </mml:msup> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">O(n^{-r})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in the Wasserstein <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -metric for all <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="r greater-than 1 slash 4"> <mml:semantics> <mml:mrow> <mml:mi>r</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>1</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>4</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">r&gt;1/4</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p greater-than normal infinity"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mi mathvariant="normal"> ∞ </mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">p&gt;\infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . Additionally, this is the first result on rates that covers certain invertible, slowly mixing maps, such as Bunimovich flowers.